W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

published:17 May 2013

views:67997

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

published:01 May 2013

views:26974

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn [or half-slope--I have changed terminology since this video was made!] instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier. #HistoryChannel
Subscribe for more from HISTORY:
http://www.youtube.com/subscription_center?add_user=historychannel
Find out more about this and other specials on our site:
http://www.history.com
Newsletter: https://www.history.com/newsletter
Website - http://www.history.com
Facebook - https://www.facebook.com/History
Twitter - https://twitter.com/history
Google+ - https://plus.google.com/+HISTORY
HISTORY SpecialsSeason 1Episode 1
THE HISTORY CHANNEL brings history's most incredible wartime feats, scientific mysteries, and turbulent periods back to life.
HISTORY®, now reaching more than 98 million homes, is the leading destination for award-winning original series and specials that connect viewers with history in an informative, immersive, and entertaining manner across all platforms. The network’s all-original programming slate features a roster of hit series, epic miniseries, and scripted event programming. Visit us at HISTORY.com for more info.

published:07 Jul 2017

views:168796

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west of Scotland, we join him for a fun journey and ask him what else is important in his life. See more at http://crazyway.tv

The 2017 Formula 1 season was one of the most thrilling in recent years. For the first time since the V6 Turbo era started, there were two Formula 1 teams with chances to fight for the Formula One World Championship. One of them was the Mercedes AMG PetronasFormula OneTeam, wich was the winner of the Formula One World Championship the last three years. The other one is the most succesful team in the Formula 1 history, the Scuderia Ferrari.
The German driver Sebastian Vettel, a four time F1 world champion with Red Bull Racing, is the man who must lead the Scuderia Ferrari to the glory, meanwhile the british driver Lewis Hamilton, a three time F1 world champion with Mclaren one time and with Mercedes the other two, is the man who must defend the Mercedes hegemony and fight for his fourth championship to try to match Vettel's titles.
Hamilton and Vettel add together seven F1 world championships and both have won multiple races. For the first time they fight for a F1 world title to discover who is the fastest man on the planet. Enjoy this epic battle full of tension, drama and thrill between two historic Formula one teams and two champion drivers. Enjoy this Clash of Champions.
Produced by FLoz - 2017
----------------------------------------------------------------------------------
📱 My Social Media
▪️ Twitter ➥ https://twitter.com/FLoz_1
▪️ Vimeo ➥ https://vimeo.com/floz
🎼 MusicComposers
▪️ Capo Productions
https://capoproductions.bandcamp.com/
https://www.youtube.com/user/CapoProductionz
▪️ DoomTillDawn Music (DTD Music)
https://soundcloud.com/mehmettorcuk
https://www.youtube.com/channel/UC3o_IDpOIA1rD1o6nEXHg4g
▪️ Kai Engel
https://www.kai-engel.com/
https://www.youtube.com/channel/UCN4bhxAz0GUg98Yuqr8hRTA
▪️ MakaiSymphony
https://soundcloud.com/makai-symphony
https://www.youtube.com/channel/UC3o_IDpOIA1rD1o6nEXHg4g
ℹ More Info
-----------------------
Photographies
-----------------------
▪️ Formula 3 Euro Serieshttp://www.fiaf3europe.com/galerien/
▪️ Getty Images
http://www.gettyimages.com
▪️ Philip Brown
http://www.philipbrownphotos.com
▪️ Rex Features
https://www.rexfeatures.com
-----------------------------------------
Team radio transcriptions
-----------------------------------------
▪️ Keith Collantine
https://twitter.com/keithcollantine
░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░
CopyrightDisclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism, comment, news reporting, teaching, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, educational or personal use tips the balance in favor of fair use.
I'm not the owner of the footage used in this video. All rights are property of Formula One Management , My Canal, Channel 4, Sky Sports F1 and Ziggo Sport. You can find the original footage and more videos in:
- https://www.formula1.com
- https://www.mycanal.fr/sport/formule1
- http://f1.channel4.com/
- http://www.skysports.com/f1
- https://www.ziggosport.nl/racing/
Thank you for enjoying this video 😉

published:22 Dec 2017

views:1160044

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

Hamilton (crater)

Hamilton is a lunarimpact crater that is located near the southeastern limb of the Moon. From the Earth this crater is viewed nearly from the edge, limiting the amount of detail that can be observed. It can also become hidden from sight due to libration, or brought into a more favorably viewing position.

This crater is situated almost due east of the lava-flooded crater Oken, near the uneven Mare Australe. To the northeast of Hamilton, along the lunar limb, is the flooded crater Gum. Less than three crater diameters to the south is the flooded walled plain Lyot.

This is a nearly circular crater, although the rim to the north is somewhat straightened. It has a well-formed edge that has not been noticeably degraded through impact erosion. There are terraces along the interior sides, particularly along the western edge (which is hidden from view from the Earth.) The interior floor is deep and uneven, with an impact feature joining the midpoint to the north-northwestern inner wall.

In the early 1990s, GO Transit provided service out of two different facilities in Hamilton: trains were routed along the CN Grimsby subdivision to the Hamilton CNR Station 1.6km to the north, and buses operated out of an older bus station at on the northern edge of Hamilton's Central Business District at John Street North and Rebecca Street. In order to better connect GO Transit service to Hamilton's CBD, improve the interface with the Hamilton Street Railway, and consolidate train and bus services at a single site, renovations were undertaken to convert the TH&B station into the Hamilton GO Centre. The new facility, designed by Garwood-Jones & Hanham Architects, opened on April 30, 1996.

Initially an agricultural service centre, Hamilton now has a growing and diverse economy and is the third fastest growing urban area in New Zealand (behind Pukekohe and Auckland). Education and research and development play an important part in Hamilton's economy, as the city is home to approximately 40,000 tertiary students and 1,000 PhD-qualified scientists.

History

Discovery (Shanice album)

Discovery is the debut studio album by American R&B/pop singer Shanice Wilson, released October 21, 1987 by A&M Records. Shanice at the time was fourteen years old with a very mature singing voice. Singer Teena Marie originally produced the majority of the album, but A&M Records felt the songs were too mature for her age. Bryan Loren was then chosen by A&M Records to produce new tracks that were used for the album. The singles "(Baby Tell Me) Can You Dance," and "No 1/2 Steppin'" were top 10 R&B hits. "The Way You Love Me," and "I'll Bet She's Got A Boyfriend" were the final singles from the album.

Discovery (music video)

Recorded in 1979 shortly after the completion of the Discovery studio album. The track listing is identical to the studio LP; each of the album's songs has its own corresponding promotional video. It received TV airings on The Blue Jean Network in 1980 among others, with releases on VHS in 1979, then later on the "Out of the Blue Tour" Live at Wembley/Discovery 1998 DVD/VHS. The video album was produced because Jeff Lynne refused to go on tour to promote the album as was customary but instead presented it in the relatively new video format. This helped launch the nascent long-form music video market. The song's videos marked the last appearance for the band's cellists.

Album information

The first track of the album, "To France", features Maggie Reilly on vocals. The second track, "Poison Arrows", seamlessly continues from the ending of "To France" and features Barry Palmer on vocals. According to Oldfield "The Lake" was inspired by his time in time in Switzerland and around Lake Geneva. The rear cover of some issues of the album are titled Discovery and The Lake.

During 1984 Oldfield and his band embarked on a Europe wide tour in promotion of the album.

2016 reissue

Discovery was re-released in a deluxe edition format on 29 January 2016, as per all previous albums which were originally released on the Virgin label.

Formula One

Formula One (also Formula 1 or F1) is the highest class of single-seatauto racing that is sanctioned by the Fédération Internationale de l'Automobile (FIA). The FIA Formula One World Championship has been the premier form of racing since the inaugural season in 1950, although other Formula One races were regularly held until 1983. The "formula", designated in the name, refers to a set of rules, to which all participants' cars must conform. The F1 season consists of a series of races, known as Grands Prix (from French, originally meaning great prizes), held throughout the world on purpose-built F1 circuits and public roads.

The results of each race are evaluated using a points system to determine two annual World Championships, one for drivers, one for constructors. The racing drivers are required to be holders of valid Super Licences, the highest class of racing licence issued by the FIA. The races are required to be held on tracks graded 1 (formerly A), the highest grade a track can receive by the FIA. Most events are held in rural locations on purpose-built tracks, but there are several events in city centres throughout the world, with the Monaco Grand Prix being the most obvious and famous example.

The rotation problem and Hamilton's discovery of quaternions I | Famous Math Problems 13a

The rotation problem and Hamilton's discovery of quaternions I | Famous Math Problems 13a

The rotation problem and Hamilton's discovery of quaternions I | Famous Math Problems 13a

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn [or half-slope--I have changed terminology since this video was made!] instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier. #HistoryChannel
Subscribe for more from HISTORY:
http://www.youtube.com/subscription_center?add_user=historychannel
Find out more about this and other specials on our site:
http://www.history.com
Newsletter: https://www.history.com/newsletter
Website - http://www.history.com
Facebook - https://www.facebook.com/History
Twitter - https://twitter.com/history
Google+ - https://plus.google.com/+HISTORY
HISTORY SpecialsSeason 1Episode 1
THE HISTORY CHANNEL brings history's most incredible wartime feats, scientific mysteries, and turbulent periods back to life.
HISTORY®, now reaching more than 98 million homes, is the leading destination for award-winning original series and specials that connect viewers with history in an informative, immersive, and entertaining manner across all platforms. The network’s all-original programming slate features a roster of hit series, epic miniseries, and scripted event programming. Visit us at HISTORY.com for more info.

4:47

Forbes Hamilton, Land Rover Discovery

Forbes Hamilton, Land Rover Discovery

Forbes Hamilton, Land Rover Discovery

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west of Scotland, we join him for a fun journey and ask him what else is important in his life. See more at http://crazyway.tv

Silver v Red - Clash of Champions

The 2017 Formula 1 season was one of the most thrilling in recent years. For the first time since the V6 Turbo era started, there were two Formula 1 teams with chances to fight for the Formula One World Championship. One of them was the Mercedes AMG PetronasFormula OneTeam, wich was the winner of the Formula One World Championship the last three years. The other one is the most succesful team in the Formula 1 history, the Scuderia Ferrari.
The German driver Sebastian Vettel, a four time F1 world champion with Red Bull Racing, is the man who must lead the Scuderia Ferrari to the glory, meanwhile the british driver Lewis Hamilton, a three time F1 world champion with Mclaren one time and with Mercedes the other two, is the man who must defend the Mercedes hegemony and fight for his fourth championship to try to match Vettel's titles.
Hamilton and Vettel add together seven F1 world championships and both have won multiple races. For the first time they fight for a F1 world title to discover who is the fastest man on the planet. Enjoy this epic battle full of tension, drama and thrill between two historic Formula one teams and two champion drivers. Enjoy this Clash of Champions.
Produced by FLoz - 2017
----------------------------------------------------------------------------------
📱 My Social Media
▪️ Twitter ➥ https://twitter.com/FLoz_1
▪️ Vimeo ➥ https://vimeo.com/floz
🎼 MusicComposers
▪️ Capo Productions
https://capoproductions.bandcamp.com/
https://www.youtube.com/user/CapoProductionz
▪️ DoomTillDawn Music (DTD Music)
https://soundcloud.com/mehmettorcuk
https://www.youtube.com/channel/UC3o_IDpOIA1rD1o6nEXHg4g
▪️ Kai Engel
https://www.kai-engel.com/
https://www.youtube.com/channel/UCN4bhxAz0GUg98Yuqr8hRTA
▪️ MakaiSymphony
https://soundcloud.com/makai-symphony
https://www.youtube.com/channel/UC3o_IDpOIA1rD1o6nEXHg4g
ℹ More Info
-----------------------
Photographies
-----------------------
▪️ Formula 3 Euro Serieshttp://www.fiaf3europe.com/galerien/
▪️ Getty Images
http://www.gettyimages.com
▪️ Philip Brown
http://www.philipbrownphotos.com
▪️ Rex Features
https://www.rexfeatures.com
-----------------------------------------
Team radio transcriptions
-----------------------------------------
▪️ Keith Collantine
https://twitter.com/keithcollantine
░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░
CopyrightDisclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism, comment, news reporting, teaching, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, educational or personal use tips the balance in favor of fair use.
I'm not the owner of the footage used in this video. All rights are property of Formula One Management , My Canal, Channel 4, Sky Sports F1 and Ziggo Sport. You can find the original footage and more videos in:
- https://www.formula1.com
- https://www.mycanal.fr/sport/formule1
- http://f1.channel4.com/
- http://www.skysports.com/f1
- https://www.ziggosport.nl/racing/
Thank you for enjoying this video 😉

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

3:24

Loading and Driving a 30 metre long Pile at the Hamilton Marine Discovery Centre

Loading and Driving a 30 metre long Pile at the Hamilton Marine Discovery Centre

Loading and Driving a 30 metre long Pile at the Hamilton Marine Discovery Centre

Hamilton, Boole and their Algebras - Professor Raymond Flood

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,800 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: https://www.facebook.com/greshamcollege
Instagram: http://www.instagram.com/greshamcollege

The rotation problem and Hamilton's discovery of quaternions I | Famous Math Problems 13a

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at ...

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors ...

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn [or half-slope--I have changed terminology since this video was made!] instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of s...

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier. #HistoryChannel
Subscribe for more from HISTORY:
http://www.youtube.com/subscription_center?add_user=historychannel
Find out more about this and other specials on our site:
http://www.history.com
Newsletter: https://www.history.com/newsletter
Website - http://www.history.com
Facebook - https://www.facebook.com/History
Twitter - https://twitter.com/history
Google+ - https://plus.google.com/+HISTORY
HISTORY SpecialsSeason 1Episode 1
THE HISTORY CHANNEL brings history's most incredible wartime feats, scientific mysteries, and turbulent periods back to life.
HISTORY®, now reaching more than 98 million homes, is the leading destination for award-winning original series and s...

published: 07 Jul 2017

Forbes Hamilton, Land Rover Discovery

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west of Scotland, we join him for a fun journey and ask him what else is important in his life. See more at http://crazyway.tv

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slow...

published: 24 Jul 2013

Loading and Driving a 30 metre long Pile at the Hamilton Marine Discovery Centre

Hamilton, Boole and their Algebras - Professor Raymond Flood

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-...

The rotation problem and Hamilton's discovery of quaternions I | Famous Math Problems 13a

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with h...

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem ...

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion o...

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn [or half-slope--I have changed terminology since this video was made!] instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn [or half-slope--I have changed terminology since this video was made!] instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier. #HistoryChannel
Subscribe for more from HISTO...

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier. #HistoryChannel
Subscribe for more from HISTORY:
http://www.youtube.com/subscription_center?add_user=historychannel
Find out more about this and other specials on our site:
http://www.history.com
Newsletter: https://www.history.com/newsletter
Website - http://www.history.com
Facebook - https://www.facebook.com/History
Twitter - https://twitter.com/history
Google+ - https://plus.google.com/+HISTORY
HISTORY SpecialsSeason 1Episode 1
THE HISTORY CHANNEL brings history's most incredible wartime feats, scientific mysteries, and turbulent periods back to life.
HISTORY®, now reaching more than 98 million homes, is the leading destination for award-winning original series and specials that connect viewers with history in an informative, immersive, and entertaining manner across all platforms. The network’s all-original programming slate features a roster of hit series, epic miniseries, and scripted event programming. Visit us at HISTORY.com for more info.

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier. #HistoryChannel
Subscribe for more from HISTORY:
http://www.youtube.com/subscription_center?add_user=historychannel
Find out more about this and other specials on our site:
http://www.history.com
Newsletter: https://www.history.com/newsletter
Website - http://www.history.com
Facebook - https://www.facebook.com/History
Twitter - https://twitter.com/history
Google+ - https://plus.google.com/+HISTORY
HISTORY SpecialsSeason 1Episode 1
THE HISTORY CHANNEL brings history's most incredible wartime feats, scientific mysteries, and turbulent periods back to life.
HISTORY®, now reaching more than 98 million homes, is the leading destination for award-winning original series and specials that connect viewers with history in an informative, immersive, and entertaining manner across all platforms. The network’s all-original programming slate features a roster of hit series, epic miniseries, and scripted event programming. Visit us at HISTORY.com for more info.

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west of Scotland, we join him for a fun journey and ask him what else is important in his life. See more at http://crazyway.tv

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west of Scotland, we join him for a fun journey and ask him what else is important in his life. See more at http://crazyway.tv

The 2017 Formula 1 season was one of the most thrilling in recent years. For the first time since the V6 Turbo era started, there were two Formula 1 teams with chances to fight for the Formula One World Championship. One of them was the Mercedes AMG PetronasFormula OneTeam, wich was the winner of the Formula One World Championship the last three years. The other one is the most succesful team in the Formula 1 history, the Scuderia Ferrari.
The German driver Sebastian Vettel, a four time F1 world champion with Red Bull Racing, is the man who must lead the Scuderia Ferrari to the glory, meanwhile the british driver Lewis Hamilton, a three time F1 world champion with Mclaren one time and with Mercedes the other two, is the man who must defend the Mercedes hegemony and fight for his fourth championship to try to match Vettel's titles.
Hamilton and Vettel add together seven F1 world championships and both have won multiple races. For the first time they fight for a F1 world title to discover who is the fastest man on the planet. Enjoy this epic battle full of tension, drama and thrill between two historic Formula one teams and two champion drivers. Enjoy this Clash of Champions.
Produced by FLoz - 2017
----------------------------------------------------------------------------------
📱 My Social Media
▪️ Twitter ➥ https://twitter.com/FLoz_1
▪️ Vimeo ➥ https://vimeo.com/floz
🎼 MusicComposers
▪️ Capo Productions
https://capoproductions.bandcamp.com/
https://www.youtube.com/user/CapoProductionz
▪️ DoomTillDawn Music (DTD Music)
https://soundcloud.com/mehmettorcuk
https://www.youtube.com/channel/UC3o_IDpOIA1rD1o6nEXHg4g
▪️ Kai Engel
https://www.kai-engel.com/
https://www.youtube.com/channel/UCN4bhxAz0GUg98Yuqr8hRTA
▪️ MakaiSymphony
https://soundcloud.com/makai-symphony
https://www.youtube.com/channel/UC3o_IDpOIA1rD1o6nEXHg4g
ℹ More Info
-----------------------
Photographies
-----------------------
▪️ Formula 3 Euro Serieshttp://www.fiaf3europe.com/galerien/
▪️ Getty Images
http://www.gettyimages.com
▪️ Philip Brown
http://www.philipbrownphotos.com
▪️ Rex Features
https://www.rexfeatures.com
-----------------------------------------
Team radio transcriptions
-----------------------------------------
▪️ Keith Collantine
https://twitter.com/keithcollantine
░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░
CopyrightDisclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism, comment, news reporting, teaching, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, educational or personal use tips the balance in favor of fair use.
I'm not the owner of the footage used in this video. All rights are property of Formula One Management , My Canal, Channel 4, Sky Sports F1 and Ziggo Sport. You can find the original footage and more videos in:
- https://www.formula1.com
- https://www.mycanal.fr/sport/formule1
- http://f1.channel4.com/
- http://www.skysports.com/f1
- https://www.ziggosport.nl/racing/
Thank you for enjoying this video 😉

The 2017 Formula 1 season was one of the most thrilling in recent years. For the first time since the V6 Turbo era started, there were two Formula 1 teams with chances to fight for the Formula One World Championship. One of them was the Mercedes AMG PetronasFormula OneTeam, wich was the winner of the Formula One World Championship the last three years. The other one is the most succesful team in the Formula 1 history, the Scuderia Ferrari.
The German driver Sebastian Vettel, a four time F1 world champion with Red Bull Racing, is the man who must lead the Scuderia Ferrari to the glory, meanwhile the british driver Lewis Hamilton, a three time F1 world champion with Mclaren one time and with Mercedes the other two, is the man who must defend the Mercedes hegemony and fight for his fourth championship to try to match Vettel's titles.
Hamilton and Vettel add together seven F1 world championships and both have won multiple races. For the first time they fight for a F1 world title to discover who is the fastest man on the planet. Enjoy this epic battle full of tension, drama and thrill between two historic Formula one teams and two champion drivers. Enjoy this Clash of Champions.
Produced by FLoz - 2017
----------------------------------------------------------------------------------
📱 My Social Media
▪️ Twitter ➥ https://twitter.com/FLoz_1
▪️ Vimeo ➥ https://vimeo.com/floz
🎼 MusicComposers
▪️ Capo Productions
https://capoproductions.bandcamp.com/
https://www.youtube.com/user/CapoProductionz
▪️ DoomTillDawn Music (DTD Music)
https://soundcloud.com/mehmettorcuk
https://www.youtube.com/channel/UC3o_IDpOIA1rD1o6nEXHg4g
▪️ Kai Engel
https://www.kai-engel.com/
https://www.youtube.com/channel/UCN4bhxAz0GUg98Yuqr8hRTA
▪️ MakaiSymphony
https://soundcloud.com/makai-symphony
https://www.youtube.com/channel/UC3o_IDpOIA1rD1o6nEXHg4g
ℹ More Info
-----------------------
Photographies
-----------------------
▪️ Formula 3 Euro Serieshttp://www.fiaf3europe.com/galerien/
▪️ Getty Images
http://www.gettyimages.com
▪️ Philip Brown
http://www.philipbrownphotos.com
▪️ Rex Features
https://www.rexfeatures.com
-----------------------------------------
Team radio transcriptions
-----------------------------------------
▪️ Keith Collantine
https://twitter.com/keithcollantine
░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░
CopyrightDisclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism, comment, news reporting, teaching, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, educational or personal use tips the balance in favor of fair use.
I'm not the owner of the footage used in this video. All rights are property of Formula One Management , My Canal, Channel 4, Sky Sports F1 and Ziggo Sport. You can find the original footage and more videos in:
- https://www.formula1.com
- https://www.mycanal.fr/sport/formule1
- http://f1.channel4.com/
- http://www.skysports.com/f1
- https://www.ziggosport.nl/racing/
Thank you for enjoying this video 😉

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discove...

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,800 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: https://www.facebook.com/greshamcollege
Instagram: http://www.instagram.com/greshamcollege

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,800 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: https://www.facebook.com/greshamcollege
Instagram: http://www.instagram.com/greshamcollege

The rotation problem and Hamilton's discovery of quaternions I | Famous Math Problems 13a

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn [or half-slope--I have changed terminology since this video was made!] instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier. #HistoryChannel
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HISTORY SpecialsSeason 1Episode 1
THE HISTORY CHANNEL brings history's most incredible wartime feats, scientific mysteries, and turbulent periods back to life.
HISTORY®, now reaching more than 98 million homes, is the leading destination for award-winning original series and specials that connect viewers with history in an informative, immersive, and entertaining manner across all platforms. The network’s all-original programming slate features a roster of hit series, epic miniseries, and scripted event programming. Visit us at HISTORY.com for more info.

Forbes Hamilton, Land Rover Discovery

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west of Scotland, we join him for a fun journey and ask him what else is important in his life. See more at http://crazyway.tv

Silver v Red - Clash of Champions

The 2017 Formula 1 season was one of the most thrilling in recent years. For the first time since the V6 Turbo era started, there were two Formula 1 teams with chances to fight for the Formula One World Championship. One of them was the Mercedes AMG PetronasFormula OneTeam, wich was the winner of the Formula One World Championship the last three years. The other one is the most succesful team in the Formula 1 history, the Scuderia Ferrari.
The German driver Sebastian Vettel, a four time F1 world champion with Red Bull Racing, is the man who must lead the Scuderia Ferrari to the glory, meanwhile the british driver Lewis Hamilton, a three time F1 world champion with Mclaren one time and with Mercedes the other two, is the man who must defend the Mercedes hegemony and fight for his fourth championship to try to match Vettel's titles.
Hamilton and Vettel add together seven F1 world championships and both have won multiple races. For the first time they fight for a F1 world title to discover who is the fastest man on the planet. Enjoy this epic battle full of tension, drama and thrill between two historic Formula one teams and two champion drivers. Enjoy this Clash of Champions.
Produced by FLoz - 2017
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I'm not the owner of the footage used in this video. All rights are property of Formula One Management , My Canal, Channel 4, Sky Sports F1 and Ziggo Sport. You can find the original footage and more videos in:
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Thank you for enjoying this video 😉

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

Hamilton (crater)

Hamilton is a lunarimpact crater that is located near the southeastern limb of the Moon. From the Earth this crater is viewed nearly from the edge, limiting the amount of detail that can be observed. It can also become hidden from sight due to libration, or brought into a more favorably viewing position.

This crater is situated almost due east of the lava-flooded crater Oken, near the uneven Mare Australe. To the northeast of Hamilton, along the lunar limb, is the flooded crater Gum. Less than three crater diameters to the south is the flooded walled plain Lyot.

This is a nearly circular crater, although the rim to the north is somewhat straightened. It has a well-formed edge that has not been noticeably degraded through impact erosion. There are terraces along the interior sides, particularly along the western edge (which is hidden from view from the Earth.) The interior floor is deep and uneven, with an impact feature joining the midpoint to the north-northwestern inner wall.